SAT & ACT Foundations

E SAT & ACT Foundations SA M PL MATHEMATICS (800) MY TUTOR Focusing on the Individual Student MYTUTOR.COM

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CONTENTS ARITHMETIC Even and Odd Integers Positive and Negative Numbers Absolute Value Other Types of Integers Divisibility and Remainders Multiples and Factors Fraction Basics Lowest Common Denominator Adding and Subtracting Fractions Multiplying and Dividing Fractions Decimals Percents Common Percent Questions Ratios Proportions Exponents Exponent Rules Radicals Sequences ALGEBRA & FUNCTIONS Algebraic Expressions Solving Equations Simultaneous Equations Equations with Radicals / Rational Expressions 6 8 10 12 14 16 18 20 22 24 26 28 0 2 4 6 8 40 42 46 48 0 2

Quadratic Equations Inequalities / Absolute Value Equations Functions Translation Word Problems Logarithms (ACT only) Imaginary Numbers (ACT only) Matrices (ACT only) COORDINATE GEOMETRY Coordinate Plane Midpoint and Distance Slope Graphing Lines Graphing Functions Function Charts Parabolas and Circles PLANE GEOMETRY Angles Vertical Angles and Parallel Lines Triangles Right Triangles Similar Triangles Area and Perimeter Circles Volume and Surface Area Trigonometry (ACT only) 4 6 8 60 62 64 66 68 72 74 76 79 82 8 88 92 9 98 101 104 107 110 11 116

STATISTICS Average Median and Mode Combinations 120 122 124 Probability 126

18 ARITHMETIC Fraction Basics A solid understanding of fractions is essential to the SAT and ACT. Many problems will require you to know how to convert and reduce fractions. Fractions are used to represent portions of a whole. Fractions also represent divisions. The fraction 4 The fraction 4 represents portions of the whole, which is made up of 4 portions. also represents divided by 4. The top part of a fraction is the numerator. This is the part or portion of the whole. The bottom part of a fraction is the denominator. This is the whole. An improper fraction has a numerator that is greater than the denominator. Mixed numbers are mixed because they have a whole number part and a fraction part. A mixed number is the sum of the whole number part and the fraction part. 1 is the same as 1 +. 8 8 The easiest way to work with mixed numbers is to convert them to improper fractions. To convert a mixed number to an improper fraction, rewrite the whole number part as a fraction with the same denominator as the fraction part, and then add that to the original fraction part. Convert 2 to an improper fraction. 2 2 + 10 + To reduce fractions, divide the numerator and denominator by a common factor. This is also called simplifying a fraction, or putting a fraction in lowest terms. 1 6 9 reduces to 2 by dividing the numerator and denominator by their common factor.

ARITHMETIC 19 TRY IT OUT 1. 4 is equal to which of the following? 10 2 2 12 12 40,,,, 0 18 100 Convert mixed numbers to improper fractions: 1. 2 8 6. 4 4 2. 16 24 is equal to which of the following? 4 6 20 2 8,,,, 6 9 28 10. Reduce 4 100 4. Reduce 6 18 PUT IT TOGETHER SAT Questions to lowest terms to lowest terms 1. If 14 + k 2 = what is the value of k? (A) 1 (B) 2 (C) (D) 4 (E) 2. The numerator of a certain fraction is x and the denominator is x + 6. If the fraction is equal to, what is the value of x? (A) (B) 6 (C) 9 (D) 10 (E) 1 7. 12 17 Convert improper fractions to mixed numbers: 8. 1 8 9. 27 10. x 2x ACT Questions 1. What number can be subtracted from both the numerator and denominator of 7 to get a value 1 equal to 1? A. 1 B. 2 C. D. 4 E. 2. The denominator of a certain fraction is more than the numerator. The fraction is equal to 2. What is the numerator of this fraction? F. G. 6 H. 7 J. 8 K. 9

28 ARITHMETIC Percents Percents appear frequently on the SAT and ACT. It s important to understand what percents are and how they relate to fractions and decimals. Percents are a way of expressing fractions with a denominator of 100. The term percent means out of one hundred. To convert a percent to a fraction, make the percent value a numerator over a denominator of 100, then simplify. 7 7% = 100 440 2 440% = = 4 100 x To convert a fraction to a percent, set up a proportion with the fraction equal to 100 and solve for x. = x % x = 100 x = 100 = x x = 60 100 To convert a percent to a decimal, divide the percent value by 100 (move the decimal point two places to the left). To convert a decimal to a percent, multiply the value by 100 (move the decimal point two places to the right). Percents, decimals, and fractions can all be used interchangeably. Complete the table of common conversions below. Percent Fraction Decimal 20% 0.1 0% 1 0.2

ARITHMETIC 29 TRY IT OUT Change the following percents to fractions and reduce to lowest terms. 1. 68% 2. 140%. 1 % Change the following fractions to percents. 4. 4. 6. 8 4 2 PUT IT TOGETHER SAT Questions 1. Which of the following is NOT equivalent to 1 4 of 60% of 80? (A) 1% of 80 (B) 1 of 48 4 (C) 60% of 20 (D) 2% of 72 (E) 12 2. Mr. Rodriguez gave a math exam with 2 problems, all of which were worth the same. Which of the following could NOT be a possible percentage of the problems that a student correctly answered? (A) 44% (B) 0% (C) 60% (D) 72% (E) 76% Change the following percents to decimals. 7. 7% 8. 160% 9..08% 10. 1 7 % 2 Change the following decimals to percents. 11. 0.6 12. 2.07 1. 0.126 14. 0. ACT Questions 1. An iceberg weighs 40,000 pounds at the beginning of the day. If it loses 0.2% of its weight in a day, how much does it weigh, in pounds, at the end of the day? A. 40,901 B. 449,800 C. 449,100 D. 446,400 E. 441,000 2. When Tuyen won a cash prize, he gave 20% to charity and had $1400 remaining. How much was the cash prize originally worth? F. $1,22 G. $1,7 H. $1,600 J. $1,680 K. $1,70

110 PLANE GEOMETRY Circles When the SAT or ACT gives you some information relating to a circle, think of what other information you can find. For example, know how to find the area when you re given the circumference, and vice versa. The radius of a circle is a line segment that starts at the center and ends on the edge of the circle. The diameter of a circle is a line segment that passes directly through the center of the circle and is twice as long as the radius. All diameters in a given circle have the same length. The circumference is the distance around the edge of the circle. Circumference = 2πr or πd A portion of the circumference is an arc. The length of an arc is a fraction of the circumference determined by the central angle. Length of arc = central angle 2πr 60 8 Radius = 8 Circumference = 2π (8) 16π Length of arc = 90 60 2π (8) 1 16π 4π 4 The area of a circle is the amount of surface inside the circle. Area = πr 2 A portion of the area, including a central angle, is called a sector. central angle 2 Area of sector = πr 60 Radius = 6 radius diameter 6 120 Area = π(6) 2 6 π Area of sector = 120 60 1 π(6)2 6π 12 π

PLANE GEOMETRY 111 TRY IT OUT Find the area and circumference of the following circles. Either the radius or diameter is given. Find the area of the sectors and length of the arcs defined by the given angles. 1. 4 9. 12 2.. 7 10 4. A circle has an area of π. What is its circumference?. A circle has an area of 64π. What is its circumference? 6. A circle has a circumference of 9π. What is its area? 7. A circle has an area of 0.2π. What is its circumference? 8. A circle has a circumference of 0.π. What is its area? 10. 11. 12. 1. 60 6 120 80 9 12 6 00

112 PLANE GEOMETRY PUT IT TOGETHER SAT Questions ACT Questions 1. In the figure above, the two circles are tangent at point A. Point B is the center of the larger circle, and AB is a diameter of the smaller circle. If AB = 6, what is the area of the shaded region? (A) 108π (B) 4π (C) 6π 6 (D) 6π 9 (E) 27π 2. In the figure above, regular hexagon ABCDEF is inscribed in a circle. If the diameter of the circle is 6, what is the length of arc EFA? (A) π (B) 2π (C) π (D) 4π (E) 6π B A C A B D F E 1. The diameter of circle X is four times the diameter of circle Y. What is the ratio of the area of circle Y to the area of circle X? A. 16:1 B. 4:1 C. 1:2 D. 1:4 E. 1:16 2. The smaller of two concentric circles has a 2- foot radius. The distance between the circles is 6 feet. Which of the following is an expression for the difference in the areas of the two circles? F. ( 6 2) 2 π G. ( 2 2 ) 6 2 π H. ( ) 2 8 2 π J. ( 2 2 ) K. ( 2 2 ) 8 2 π 8 6 π